1. JAIRO SINOVA Research fueled by: NERC ICNM 2007, Istanbul, Turkey July 25 th 2007 Anomalous and Spin-Hall effects in mesoscopic systems

2. Branislav Nikolic U. of Delaware Allan MacDonald U of Texas Tomas Jungwirth Inst. of Phys. ASCR U. of Nottingham Joerg Wunderlich Cambridge-Hitachi Laurens Molenkamp Wuerzburg Kentaro Nomura U. Of Texas Ewelina Hankiewicz U. of Missouri Texas A&M U. Mario Borunda Texas A&M U. Nikolai Sinitsyn Texas A&M U. U. of Texas Other collaborators: Bernd Kästner, Satofumi Souma, Liviu Zarbo, Dimitri Culcer, Qian Niu, S-Q Shen, Brian Gallagher, Tom Fox, Richard Campton, Winfried Teizer, Artem Abanov Alexey Kovalev Texas A&M U.

3. OUTLINE The three spintronic Hall effects The three spintronic Hall effects Anomalous Hall effect and Spin Hall effect Anomalous Hall effect and Spin Hall effect AHE phenomenology and its long history AHE phenomenology and its long history Three contributions to the AHE Three contributions to the AHE Microscopic approach: focus on the intrinsic AHE Microscopic approach: focus on the intrinsic AHE Application to the SHE Application to the SHE SHE in Rashba systems: a lesson from the past SHE in Rashba systems: a lesson from the past Recent experimental results Recent experimental results Spin Hall spin accumulation: bulk and mesoscopic regime Spin Hall spin accumulation: bulk and mesoscopic regime Mesoscopic spin Hall effect: non-equilibrium Green’s function formalism and recent experiments Mesoscopic spin Hall effect: non-equilibrium Green’s function formalism and recent experiments Summary Summary

4. The spintronics Hall effects AHE SHE charge current gives spin current polarized charge current gives charge-spin current SHE -1 spin current gives charge current

5. Anomalous Hall transport Commonalities: Spin-orbit coupling is the key Same basic (semiclassical) mechanisms Differences: Charge-current (AHE) well define, spin current (SHE) is not Exchange field present (AHE) vs. non- exchange field present (SHE -1 ) Difficulties: Difficult to deal systematically with off-diagonal transport in multi-band system Large SO coupling makes important length scales hard to pick Conflicting results of supposedly equivalent theories The Hall conductivities tend to be small

6. Spin-orbit coupling interaction (one of the few echoes of relativistic physics in the solid state) Ingredients: -“Impurity” potential V(r) - Motion of an electron Produces an electric field In the rest frame of an electron the electric field generates and effective magnetic field This gives an effective interaction with the electron’s magnetic moment CONSEQUENCES If part of the full Hamiltonian quantization axis of the spin now depends on the momentum of the electron !! If treated as scattering the electron gets scattered to the left or to the right depending on its spin!!

7. Anomalous Hall effect: where things started, the long debate Simple electrical measurement of magnetization like-spin Spin-orbit coupling “force” deflects like-spin particles I _ F SO _ _ _ majority minority V InMnAs controversial theoretically: semiclassical theory identifies three contributions (intrinsic deflection, skew scattering, side jump scattering)

8. (thanks to P. Bruno– CESAM talk) A history of controversy

9. Intrinsic deflection Electrons have an “anomalous” velocity perpendicular to the electric field related to their Berry’s phase curvature which is nonzero when they have spin-orbit coupling. Movie created by Mario Borunda Electrons deflect to the right or to the left as they are accelerated by an electric field ONLY because of the spin-orbit coupling in the periodic potential (electronics structure)

10. Skew scattering Movie created by Mario Borunda Asymmetric scattering due to the spin-orbit coupling of the electron or the impurity. This is also known as Mott scattering used to polarize beams of particles in accelerators.

11. Side-jump scattering Movie created by Mario Borunda Related to the intrinsic effect: analogy to refraction from an imbedded medium Electrons deflect first to one side due to the field created by the impurity and deflect back when they leave the impurity since the field is opposite resulting in a side step.

12. THE THREE CONTRIBUTIONS TO THE AHE: MICROSCOPIC KUBO APPROACH Skew scattering Side-jump scattering Intrinsic AHE Skew σ H Skew  (  skew ) -1  2~ σ 0 S where S = Q(k,p)/Q(p,k) – 1~ V 0 Im[ ] Vertex Corrections  σ Intrinsic Intrinsic  σ 0 /ε F  n, q m, p n’, k n, q n’  n, q =  -1 / 0 Averaging procedures: = 0 

13. FOCUS ON INTRINSIC AHE: semiclassical and Kubo K. Ohgushi, et al PRB 62, R6065 (2000); T. Jungwirth et al PRL 88, 7208 (2002); T. Jungwirth et al. Appl. Phys. Lett. 83, 320 (2003); M. Onoda et al J. Phys. Soc. Jpn. 71, 19 (2002); Z. Fang, et al, Science 302, 92 (2003). Semiclassical approach in the “clean limit” Kubo: n, q n’  n, q STRATEGY: compute this contribution in strongly SO coupled ferromagnets and compare to experimental results, does it work?

14. Success of intrinsic AHE approach DMS systems (Jungwirth et al PRL 2002) Fe (Yao et al PRL 04) Layered 2D ferromagnets such as SrRuO3 and pyrochlore ferromagnets [Onoda and Nagaosa, J. Phys. Soc. Jap. 71, 19 (2001),Taguchi et al., Science 291, 2573 (2001), Fang et al Science 302, 92 (2003), Shindou and Nagaosa, Phys. Rev. Lett. 87, 116801 (2001)] Colossal magnetoresistance of manganites, Ye et~al Phys. Rev. Lett. 83, 3737 (1999). Ferromagnetic Spinel CuCrSeBr: Wei-Lee et al, Science (2004) Berry’s phase based AHE effect is quantitative-successful in many instances BUT still not a theory that treats systematically intrinsic and extrinsic contribution in an equal footing. Experiment  AH  1000 (  cm) -1 Theroy  AH  750 (  cm) -1

15. Spin Hall effect like-spin Take now a PARAMAGNET instead of a FERROMAGNET: Spin-orbit coupling “force” deflects like-spin particles I _ F SO _ _ _ V=0 non-magnetic Spin-current generation in non-magnetic systems without applying external magnetic fields Spin accumulation without charge accumulation excludes simple electrical detection Carriers with same charge but opposite spin are deflected by the spin-orbit coupling to opposite sides.

16. Spin Hall Effect (Dyaknov and Perel) Interband Coherent Response  (E F  ) 0 Occupation # Response `Skew Scattering‘  (e 2 /h) k F (E F  ) 1 X `Skewness’ [Hirsch, S.F. Zhang] Intrinsic `Berry Phase’  (e 2 /h) k F  [Murakami et al, Sinova et al] Influence of Disorder `Side Jump’’ [Inoue et al, Misckenko et al, Chalaev et al.] Paramagnets

17. INTRINSIC SPIN-HALL EFFECT: INTRINSIC SPIN-HALL EFFECT: Murakami et al Science 2003 (cond-mat/0308167) Sinova et al PRL 2004 (cont-mat/0307663) as there is an intrinsic AHE (e.g. Diluted magnetic semiconductors), there should be an intrinsic spin-Hall effect!!! Inversion symmetry  no R-SO Broken inversion symmetry  R-SO Bychkov and Rashba (1984) (differences: spin is a non-conserved quantity, define spin current as the gradient term of the continuity equation. Spin- Hall conductivity: linear response of this operator) n, q n’  n, q

18. SHE conductivity: Kubo formalism perturbation theory Skew  σ 0 S Vertex Corrections  σ Intrinsic Intrinsic  σ 0 /ε F  n, q n’  n, q = j = -e v = j z = {v,s z }

19. Disorder effects: beyond the finite lifetime approximation for Rashba 2DEG Question: Are there any other major effects beyond the finite life time broadening? Does side jump contribute significantly? Ladder partial sum vertex correction: Inoue et al PRB 04 Raimondi et al PRB 04 Mishchenko et al PRL 04 Loss et al, PRB 05 the vertex corrections are zero for 3D hole systems (Murakami 04) and 2DHG (Bernevig and Zhang 05); verified numerically by Normura et al PRB 2006 n, q n’  n, q + +…=0 For the Rashba example the side jump contribution cancels the intrinsic contribution!!

20. First experimental observations at the end of 2004 Wunderlich, Kästner, Sinova, Jungwirth, cond-mat/0410295 PRL 05 Experimental observation of the spin-Hall effect in a two dimensional spin-orbit coupled semiconductor system Co-planar spin LED in GaAs 2D hole gas: ~1% polarization Kato, Myars, Gossard, Awschalom, Science Nov 04 Observation of the spin Hall effect bulk in semiconductors Local Kerr effect in n-type GaAs and InGaAs: ~0.03% polarization (weaker SO-coupling, stronger disorder)

21. OTHER RECENT EXPERIMENTS “demonstrate that the observed spin accumulation is due to a transverse bulk electron spin current” Sih et al, Nature 05, PRL 05 Valenzuela and Tinkham cond- mat/0605423, Nature 06 Transport observation of the SHE by spin injection!! Saitoh et al APL 06

22. The new challenge: understanding spin accumulation Spin is not conserved; analogy with e-h system Burkov et al. PRB 70 (2004) Spin diffusion length Quasi-equilibrium Parallel conduction Spin Accumulation – Weak SO

23. Spin Accumulation – Strong SO Mean Free Path? Spin Precession Length ?

24. SPIN ACCUMULATION IN 2DHG: EXACT DIAGONALIZATION STUDIES  so >>ħ/  Width>>mean free path Nomura, Wundrelich et al PRB 06 Key length: spin precession length!! Independent of  !!

25. 1.5  m channel n n p y x z LED1 2 10  m channel SHE experiment in GaAs/AlGaAs 2DHG - shows the basic SHE symmetries - edge polarizations can be separated over large distances with no significant effect on the magnitude - 1-2% polarization over detection length of ~100nm consistent with theory prediction (8% over 10nm accumulation length) Wunderlich, Kaestner, Sinova, Jungwirth, Phys. Rev. Lett. '05 Nomura, Wunderlich, Sinova, Kaestner, MacDonald, Jungwirth, Phys. Rev. B '06

26. Non-equilibrium Green’s function formalism (Keldysh-LB) Advantages: No worries about spin-current definition. Defined in leads where SO=0 Well established formalism valid in linear and nonlinear regime Easy to see what is going on locally Fermi surface transport SHE in the mesoscopic regime

27. Nonequilibrium Spin Hall Accumulation in Rashba 2DEG  Spin density (Landauer –Keldysh) : +eV/2 -eV/2 eV=0 Y. K. Kato, R. C. Myers, A. C. Gossard, and D.D. Awschalom, Science 306, 1910 (2004). PRL 95, 046601 (2005)

28. band structure D.J. Chadi et al. PRB, 3058 (1972) fundamental energy gap semi-metal or semiconductor HgTe

29. Barrier QW VBO = 570 meV HgTe-Quantum Well Structures

30. Typ-III QW VBO = 570 meV HgCdTe HgTe HgCdTe HH1 E1 QW < 55 Å     HgTe invertertednormal band structure conduction band valence band HgTe-Quantum Well Structures

31. High Electron Mobility

32. H-bar for detection of Spin-Hall-Effect (electrical detection through inverse SHE) E.M. Hankiewicz et al., PRB 70, R241301 (2004)

33. Actual gated H-bar sample HgTe-QW  R = 5-15 meV 5  m ohmic Contacts Gate- Contact

34. First Data HgTe-QW  R = 5-15 meV Signal due to depletion?

35. Results... Symmetric HgTe-QW  R = 0-5 meV Signal less than 10 -4 Sample is diffusive: Vertex correction kills SHE (J. Inoue et al., Phys. Rev. B 70, 041303 (R) (2004)).

36. New (smaller) sample 1  m 200 nm sample layout

37. SHE-Measurement no signal in the n-conducting regime strong increase of the signal in the p-conducting regime, with pronounced features insulating p-conducting n-conducting

38. Mesoscopic electron SHE L L/6 L/2 calculated voltage signal for electrons (Hankiewicz and Sinova)

39. Mesoscopic hole SHE L calculated voltage signal (Hankiweicz, Sinova, & Molenkamp) L L/ 6 L/2

40. Scaling of H-samples with the system size L L/6 Oscillatory character of voltage difference with the system size.

41. Extrinsic (1971,1999) and Intrinsic (2003) SHE predicted and observed (2004): back to the beginning on a higher level Extrinsic + intrinsic AHE in graphene: two approaches with the same answer SUMMARY Optical detection of current-induced polarization photoluminescence (bulk and edge 2DHG) Kerr/Faraday rotation (3D bulk and edge, 2DEG) Transport detection of the mesoscopic SHE in semiconducting systems: HgTe preliminary results agree with theoretical calculations

43. Experimental achievements Experimental (and experiment modeling) challenges: Photoluminescence cross section edge electric field vs. SHE induced spin accumulation free vs. defect bound recombination spin accumulation vs. repopulation angle-dependent luminescence (top vs. side emission) hot electron theory of extrinsic experiments Optical detection of current-induced polarization photoluminescence (bulk and edge 2DHG) Kerr/Faraday rotation (3D bulk and edge, 2DEG) Transport detection of the SHE General edge electric field (Edelstein) vs. SHE induced spin accumulation SHE detection at finite frequencies detection of the effect in the “clean” limit WHERE WE ARE GOING (EXPERIMENTS)

44. INTRINSIC+EXTRINSIC: STILL CONTROVERSIAL! AHE in Rashba systems with disorder: Dugaev et al PRB 05 Dugaev et al PRB 05 Sinitsyn et al PRB 05 Sinitsyn et al PRB 05 Inoue et al (PRL 06) Inoue et al (PRL 06) Onoda et al (PRL 06) Onoda et al (PRL 06) Borunda et al (cond-mat 07) All are done using same or equivalent linear response formulation–different or not obviously equivalent answers!!! The only way to create consensus is to show (IN DETAIL) agreement between the different equivalent linear response theories both in AHE and SHE

45. Need to match the Kubo to the Boltzmann Need to match the Kubo to the Boltzmann Kubo: systematic formalism Kubo: systematic formalism Botzmann: easy physical interpretation of different contributions Botzmann: easy physical interpretation of different contributions Connecting Microscopic and Semiclassical approach Sinitsyn et al PRL 06, PRB 06

46. Semiclassical Boltzmann equation Golden rule: J. Smit (1956): Skew Scattering In metallic regime: Kubo-Streda formula summary

47. Golden Rule: Coordinate shift: Modified Boltzmann Equation: Berry curvature: velocity: current: Semiclassical approach II: Sinitsyn et al PRB 06 Sinitsyn et al PRB 06

48. Armchair edge Zigzag edge EFEF Some success in graphene

49. In metallic regime: Kubo-Streda formula: Single K-band with spin up Sinitsyn et al PRL 06, PRB 06 SAME RESULT OBTAINED USING BOLTMANN!!!

50. Comparing Boltzmann to Kubo in the chiral basis

51. Intrinsic deflection Electrons have an “anomalous” velocity perpendicular to the electric field related to their Berry’s phase curvature which is nonzero when they have spin-orbit coupling. Electrons deflect to the right or to the left as they are accelerated by an electric field ONLY because of the spin-orbit coupling in the periodic potential (electronics structure) E Electrons deflect first to one side due to the field created by the impurity and deflect back when they leave the impurity since the field is opposite resulting in a side step. Related to the intrinsic effect: analogy to refraction from an imbedded medium Side jump scattering Skew scattering Asymmetric scattering due to the spin- orbit coupling of the electron or the impurity. This is also known as Mott scattering used to polarize beams of particles in accelerators.